Question

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in each of the following:
(i) f(x) = x³ – 6x² + 11x – 6, g(x) = x² + x + 1
(ii) f(x) = 10x⁴ + 17x³ – 62x² + 30x – 3, g(x) = 2x² + 7x + 1
(iii) f(x) = 4x³ + 8x+ 8x² + 7, g(x) = 2x² – x + 1
(iv) f(x) = 15x³ – 20x² + 13x – 12, g(x) = 2– 2x +x²

Answer 1

(i) quotient = ax + b = x-7

remainder = 17x + 1

Step-by-step explanation:

(i) f(x) = x³ – 6x² + 11x – 6, g(x) = x² + x + 1

f(x) is of the power 3 and g(x) is of the power 2.

So quotient will be of degree 3-2 and remainder will be of power less than 2.

Let q(x) = ax + b

r(x) = cx +d

Using division algorithm, we get:

f(x) = g(x) * q(x) + r(x)

x³ – 6x² + 11x – 6 = (x² + x + 1)(ax + b) + (cx +d)

x³ – 6x² + 11x – 6 = ax³ + (a+b)x² + (a+b+c)x  + (b+d)

Equating the coefficients, we get:

x³ = ax³, therefore a = 1

-6x² = (a+b)x², so -6 = a + b = 1+b, therefore b = -7

11x = (a+b+c)x , 11 = 1 -7 +c , therefore c = 17

-6 = b+d , -6 = -7 +d , therefore d = 1

So the quotient will be q(x) = ax + b = x-7

the remainder will be cx + d = 17x + 1

Please solve the rest in the same method.